When I find an app that works, I tend to stick with it until it starts pissing me off. The Apple Podcast app worked perfectly until iOS11 when its basic functionality and ease of use was ruined. The interface has become so distorted that it's impossible to even listen to podcasts in sequential order without having to muss and fuss with it. I tolerated this for months until I finally gave up and started the hunt for something new. And, at last, I have found it! The app is called Overcast. It's simple to use and delivers exactly what you need plus a couple of extras.
Download for free here: Overcast